I'm going over past exam papers for a security module I'm taking at university and there is a question which I can't give a great answer for.

As a security manager in a bank you are asked to implemented an
"append-only" log system in Unix for all transactions in your bank.
Give five security issues you are going to address and discuss a
proper countermeasure for each.

To make it more in the spirit of this site... What security issues are there when it comes to implementing append-only log files in Unix and what would countermeasures to those issues be?

The two answers currently here are great. You should also consider the issue of how you might archive old log files without introducing a window of attack, as well as what policies you should put in place when destroying very old archives.
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PolynomialMay 21 '12 at 14:49

How will we enforce/ensure that the log can only be appended to, and no prior entries can be deleted/removed/modified?

Who/what will have authority to append items to the log?

Who/what will have authority to read the log?

What do we need to do to ensure that confidential information is not stored in the log, or else is adequately protected?

How will this stand up in court, if the logs ever need to be used as evidence in a legal dispute?

What information/events should I log?

How will we secure the information to be logged while it is in transit from the system generating the log to the system storing the log records?

How will we secure the log information on storage?

How will the log records be used? How can they be used to detect attacks/fraud/anomalies? What tools/techniques will be used to analyze the logs?

For techniques you can use to address these security issues, you might start by reading the following questions on this site (they should cover a good part of the space, and if there is anything specific missing, you can ask another more specific question separately):